We present ALMA ∼0 02 resolution observations of the nucleus of the nearby (∼14 Mpc) type 2 active galactic nucleus NGC 1068 at HCN/HCO+/HNC J = 3-2 lines, as well as at their 13C isotopologue and vibrationally excited lines, to scrutinize the morphological, dynamical, chemical, and physical properties of dense molecular gas in the putative dusty molecular torus around a mass-accreting supermassive black hole. We confirm almost east-west-oriented dense molecular gas emission both morphologically and dynamically, which we regard as coming from the torus. Bright emission is compact ( 3 pc), and low-surface-brightness emission extends out to 5-7 pc. These dense molecular gas properties are not symmetric between the eastern and western torus. The HCN J = 3-2 emission is stronger than the HCO+ J = 3-2 emission within the ∼7 pc torus region, with an estimated dense molecular mass of (0.4-1.0) × 106 M . We interpret that HCN abundance is enhanced in the torus. We detect signatures of outflowing dense molecular gas and a vibrationally excited HCN J = 3-2 line. Finally, we find that in the innermost ( 1 pc) part of the torus, the dense molecular line rotation velocity, relative to the systemic velocity, is the opposite of that in the outer ( 2 pc) part, in both the eastern and western torus. We prefer a scenario of counter-rotating dense molecular gas with innermost almost Keplerian rotation and outer slowly rotating (far below Keplerian) components. Our high-spatial-resolution dense molecular line data reveal that torus properties of NGC 1068 are much more complicated than the simple axisymmetrically rotating torus picture in the classical active galactic nucleus unification paradigm.